Battery dying after sitting overnight

My 2002 Accord battery has died 3 or 4 times in the last two weeks. It only dies after it has been sitting overnight or longer. The original battery was still in it the first time, so we went ahead and replaced that since it was 10 years old. It died again the next day - took it to a shop and the guy replaced the alternator after it tested bad.
It ran okay for 4 days, then died again. We took it back to him, and he found that there is a fan that will run randomly, even without the keys in the ignition. He said it was the AC, but I'm guessing it was the cooling fan. Is this an easy fix or could it be a number of difficult and expensive things to fix?
Also, I got the car back today, and now the AC will not cool. It puts out cool but not cold air - you can hear it cut on but the air never gets cold. There has never been a problem with the AC until today.